Spectral Methods for Time-Dependent Problems

A 2007 graduate text on spectral methods with applications in fluid dynamics and engineering.

Jan S. Hesthaven (Author), Sigal Gottlieb (Author), David Gottlieb (Author)

9780521792110, Cambridge University Press

Hardback, published 11 January 2007

284 pages, 50 b/w illus.
22.9 x 15.2 x 1.9 cm, 0.59 kg

'The book is excellent and will be valuable for post-graduate students, researchers and scientists working in applied sciences and mainly in the numerical analysis of time-dependent problems. The thoroughness of the exposition, the clarity of the mathematical techniques and the variety of the problems and theoretical results that are presented and rigorously analyzed make this book a primary reference in the advanced numerical analysis of partial differential equations.' Mathematical Reviews

Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Introduction
1. From local to global approximation
2. Trigonometric polynomial approximation
3. Fourier spectral methods
4. Orthogonal polynomials
5. Polynomial expansions
6. Polynomial approximations theory for smooth functions
7. Polynomial spectral methods
8. Stability of polynomial spectral methods
9. Spectral methods for non-smooth problems
10. Discrete stability and time integration
11. Computational aspects
12. Spectral methods on general grids
Bibliography.

Subject Areas: Topology [PBP], Geometry [PBM], Numerical analysis [PBKS], Differential calculus & equations [PBKJ]